**Ion Trapping in Dusty Plasmas**

J. Goree^{ a)}

Max-Planck-Institut für extraterrestrische Physik, 8046 Garching, Germany

Motivated by space and laboratory applications, a theory is presented for ions trapped by a charged dust grain in a plasma. The grain's attractive Debye sphere confines ions after a collision. They shield the grain's considerable electric charge from external fields. Their number Ntrap is determined by a balance of the capture and loss rates. At steady state, Ntrap is independent of the collisional mean-free-path and increases with the plasma ion density. Because of the density dependence, trapping is significant in laboratory plasmas but not in comparatively less dense space plasmas.

^{a) }Permanent address: Department of Physics and Astronomy, The
University of Iowa,

Iowa City, Iowa 52242

PACS: 52.20.Dq, 52.25.Tx, 52.25.Vy

Dusty plasmas are low-temperature ionized gases that contain particulates in
the size range 10 nm to 100 um. These dust grains collect ions and
electrons, and acquire an electric charge QD , which can be thousands of
electron charges.^{1,2} Charged grains can modify and even dominate
wave propagation,^{2,3} wave scattering,^{4} ionization
balance,^{5} shock propagation,^{5} and gradient and
velocity-space-driven instabilities.^{6,7} In space and
astrophysics,^{8} theories have been developed to explain observations
in planetary rings,^{2} interstellar clouds,^{5} Earth's
noctilucent clouds,^{9} and spacecraft-ionosphere
interactions.^{10} Laboratory dusty plasma experiments have been
performed only recently; they reveal that charged particulates can be confined
for hours in a discharge.^{11-13} This confinement leads to a serious
contamination problem in the plasma-aided manufacturing of
semiconductors,^{11-12} while it offers novel research opportunities
for basic plasma physics.^{14}

The starting point for every theory of dusty plasmas is a charging model. Now
I summarize a standard model. Here and throughout this letter I treat only the
most common case: *a* << [[lambda]]D
<< [[lambda]]mfp , where *a* is the grain radius, [[lambda]]D
the Debye length, and [[lambda]]mfp the collisional mean-free-path. The
charge QD is related to the grain's floating potential [[phi]]f by QD = C
[[phi]]f , where C = 4 [[pi]] [[epsilon]]0 *a* is the grain's
capacitance. A grain collects ion and electron currents from the
plasma,^{1,2} and these currents J[[sigma]] depend on [[phi]]f .
They add up to determine the charge, dQD / dt = [[Sigma]] J[[sigma]]
([[phi]]f) . Setting this sum to zero yields the steady-state floating
potential [[phi]]f and charge QD . Due to the low electron mass, [[phi]]f and
QD are always be negative, unless the grain emits electrons. A sphere in a
thermalized hydrogen plasma float to
, where T is the ion and electron temperature.^{15} This yields a
charge QD = -1.7 x 10^{4} e , for example, if kT = 1 eV
and a = 10 um. The actual potential and charge are somewhat less negative
than this, if ions drift with respect to the grain faster than the ion thermal
speed.^{1}

This standard charging model ignores particles that become trapped in the Debye sphere surrounding a charged grain. Because the grain has a large negative charge, it can easily hold one or more positive ions in confined orbits, as shown in Fig. 1. Trapped ions are important because they, unlike untrapped ions, shield the grain's charge from external electromagnetic fields. This is similar to the way that electrons bound in an atom cancel the charge of the nucleus within. Electromagnetic forces on the grain are thereby reduced, and other forces, such as gravity and gas drag, become more significant. The larger the number of trapped ions, Ntrap, the more effective the shielding.

The physics of trapped ions was presented earlier in theories for
probes^{16} and spacecraft,^{17} but never developed to a
useful end. Consequently theorists have neglected trapping in computing the
forces acting on dust in a plasma, without knowing knowing the validity of this
assumption. To start, it would help to have a criterion to answer the
question: when is it valid to neglect trapping? Such a criterion is
Ntrap < 1. This requires a method of computing Ntrap , which is
developed below.

To compute Ntrap , consider a positive ion, with charge qi , mass mi and angular momentum M, moving toward a negatively charged grain. The ion's total energy is , where . An ion is trapped if it is in a potential well, i.e., if it satisfies three requirements: its Ueff has a minimum, Ei is less than the maximum of Ueff found at nonzero r, and the orbit does not strike the grain. An untrapped ion can become trapped only if it undergoes a collision, scattering into a favorable direction and losing enough energy.

The rate of ion capture is

, (1)

where is the incident ion flux. Here is a capture cross section

, (2)

and Pcap(p) is the probability for an incoming ion at impact parameter p to
become captured due to a collision. Note that Pcap and
are [[proportional]] 1^{ }/^{ }[[lambda]]mfp .

What sort of collisions must be considered? Only those that result in slower ions will lead to capture. For neutral atoms and ions that are alike, charge-exchange and elastic collisions have comparable cross sections. But charge exchange yields the most ion energy loss, provided that kTgas << Ei .

The loss process is also collisional. Eventually a collision will scatter an ion enough that it becomes untrapped, either escaping to infinity or striking the grain. The loss rate is

, (3)

where [[tau]]conf is a confinement time. To make an estimate valid for isotropic scattering, it is useful to know that only one or two collisions are needed to detrap an ion; thus [[tau]]conf ~ [[lambda]]mfp / vtrap , where vtrap is a typical speed of trapped ions. A more precise value requires numerical evaluation, as done below.

A steady state is achieved when the capture and loss rates balance, . This and Eqs. (1) and (3) yield:

. (4)

This is a mean value for the number of trapped ions; the instantaneous number fluctuates in time.

Two conclusions about the physics of trapping can be made from Eq. (4). Firstly, the number Ntrap of collisionally-trapped ions is independent of the collisional mean-free-path, because the [[lambda]]mfp dependances of and in Eq. (4) cancel. Provided that charge exchange is the dominant collision process, it does not matter how low the gas density is. Secondly, the dependence on [[Gamma]]i means that the number of trapped ions is [[proportional]] ni . A grain will trap more ions if it is in a denser plasma.

I have implemented the physical model, described above, as a single-particle Monte Carlo code. The simulation has two stages. First, an incoming ion with Ei > 0 is directed toward the dust grain, at an impact parameter p . The equation of motion, , is integrated in three dimensions using a fourth-order Runge-Kutta method. To determine if a collision took place in a time step [[Delta]]t, the probability is compared to a random number. The ion orbit is followed until it either undergoes a collision or escapes from the simulation volume, a sphere of radius 5 [[lambda]]D . At a charge-exchange collision site, a new ion is born, and its initial velocity is chosen randomly from the gas thermal distribution. It is evaluated against the three requirements to see if its orbit is trapped. This is repeated for 200 to 1000 test particles, all with the same p. The capture probability Pcap(p) needed for Eq. (2) is then calculated; it is the fraction of incoming ions that yielded a trapped ion. This first stage is repeated for 40 impact parameters. Computing the integral in Eq. (2) then yields .

In the second stage, the new trapped ion orbits are followed one at a time by integrating the equation of motion. They are followed until, due to collisions, they either strike the grain or escape the simulation space. A histogram of the trapped orbit lifetime is prepared, and it is fit to a decaying exponential, , based on Eq. (3). This procedure yields the confinement time .

I make the following simplifying assumptions. First, only charge-exchange collisions are included, and second, their cross section is constant (over a decade range of energy). Third, the incoming ions are monoenergetic; this is appropriate where Ei >> Ti . Fourth, the neutral gas does not drift with respect to the grain. Fifth, the grain is spherical. Finally, the electric potential is

, (5)

which neglects any trapped orbits or wake effects.^{1,18} This
prescribed electric potential does not evolve in time as it would in a
self-consistent particle simulation. Equation (5) is valid only if: (a) the
number of trapped ions is small, Ntrap << QD / e , which may be
verified from the results, (b) the Debye spheres of two grains do not
overlap,^{19,20} (c) [[lambda]]D is much smaller than the ion
gyroradius, so that any magnetic field may be ignored, and (d) there is no
external potential drop across the Debye sphere, as found for example in the
electrode sheaths of a high-voltage laboratory discharge. In the latter case,
I speculate that a grain with a cloud of trapped ions will lose those ions when
it approaches a sheath; they will probably be detrapped when their orbits cross
into the sheath.

The simulation was performed for hydrogen, with the parameters: *a* = 10
um, [[lambda]]D = 100 um, [[lambda]]mfp = 3 mm, Ei = 1 eV, Tgas = 406 K
(kTgas = 0.035 eV). These parameters satisfy the conditions *a* <<
[[lambda]]D << [[lambda]]mfp and kTgas << Ei . Results are shown
in Figs. 2 - 3. The capture cross section
in Fig. 2 increases with the floating potential, since [[phi]]f is attractive.
The confinement time
in Fig. 2 diminishes with [[phi]]f , because
[[tau]]conf ~ [[lambda]]mfp / vtrap , and ions trapped in a shallower
potential have a lower velocity vtrap . (This result owes in part to
assuming a collision cross section that is independent of energy.) Figure 3
shows the final result, Ntrap / [[Gamma]]i as a function of floating potential,
which is useful in two ways.

Firstly, Fig. 3 can be used to compute Ntrap , provided that the ion flux is low enough that Ntrap << QD / e . For higher fluxes this will not be satisfied; Ntrap will no longer increase linearly with [[Gamma]]i , but rather saturate at QD / e . A model with a self-consistent electric potential would be required to treat the saturation regime.

Secondly, Fig. 3 allows an evaluation of the criterion Ntrap < 1,
for ion trapping to be negligible. For example, Fig. 3 shows that Ntrap /
[[Gamma]]i = 10^{-15} m^{2} s for [[phi]]f = -0.5 Volt. If
ions are streaming past the grain at vi = 10^{3} m/s , then Ntrap will
be unity for ni = 10^{12} m^{-3}. In space plasmas ni
<< 10^{12} m^{-3} , while in laboratory discharges
ni >> 10^{12} m^{-3} . This shows that ion trapping
is negligible in space physics problems, a finding that is a confidence-builder
for extant space dusty-plasma models that neglect ion trapping. But trapping
in laboratory plasmas is so strong that it is often in the saturation regime.
Previous calculations^{14,21} of electric forces acting on grains in
laboratory plasmas have ignored trapping; they are probably invalid except for
regions within a Debye length of a strong sheath.

The results for Ntrap presented in Fig. 3 were computed for particular values
of several parameters: *a* , [[lambda]]D , [[lambda]]mfp , Ei , Tgas
and ion mass. The results can be extended to other parameter values by using
the following physics. The results are independent of [[lambda]]mfp provided
that [[lambda]]mfp >> [[lambda]]D . Scaling all the relevant
energies (Ei , kTgas , and e [[phi]]f ) by the same factor will leave Ntrap
unaffected. The results should not depend strongly on the incoming ion energy
Ei , since it affects only the spatial distribution of collision sites, which
is always fairly uniform.

In summary, negatively charged dust grains in a plasma can trap positive ions in confined orbits, shielding the grain from external electromagnetic fields. The number of trapped ions is determined by a balance between collisional trapping and detrapping. A method of computing Ntrap was developed here, using a non-self-consistent electric potential, valid if Ntrap is small compared to the number of electron charges on the grain. A theory with a self-consistent electric potential still needs to be developed to compute Ntrap when it is large. I find that ion trapping is very significant in laboratory plasmas, but negligible in most space plasmas.

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**FIGURE CAPTIONS**

FIG. 1. Trapped hydrogen ion orbiting around a negatively-charged dust grain. Parameters are Ei = -0.1 eV, [[phi]]f = -2.5 V, and others listed for Fig. 2. Ueff (r) is shown in the inset.

FIG. 2. Monte Carlo simulation results for captured hydrogen ions, born by
charge exchange. The capture cross section
is computed from Eq. (2). Newly captured ions were followed until they were
detrapped due to further collisions, yielding the confinement time
. Parameters are: *a* = 10 um , [[lambda]]D = 100 um , [[lambda]]mfp=
3 mm , Ei = 1 eV. Error bars arise from counting statistics.

FIG. 3 Number of trapped ions Ntrap per unit of incoming ion flux, . These results from Eq. (4) are valid if . Unlike Fig. 2, this data is independent of [[lambda]]mfp .